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arXiv:0903.1664v2 [hep-lat] 9 Nov 2009
DESY 09-029
Semileptonic form factors D → π,K and
B → π,K from a fine lattice
A. Al-Haydari,a
G. N. Lacagnina,c
A. Ali Khan,a
M. Panero,b,d
V. M. Braun,b
A. Sch¨ aferb
S. Collins,b
and G. Schierholzb,e
M. G¨ ockeler,b
aDepartment of Physics, Faculty of Science, Taiz University, Taiz, Yemen
bInstitute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
cINFN, Sezione di Milano, 20133 Milano, Italy
dInstitute for Theoretical Physics, ETH Z¨ urich, 8093 Z¨ urich, Switzerland
eDeutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany
QCDSF Collaboration
Abstract
We extract the form factors relevant for semileptonic decays of D and B mesons from
a relativistic computation on a fine lattice in the quenched approximation. The lattice
spacing is a = 0.04 fm (corresponding to a−1= 4.97 GeV), which allows us to run very
close to the physical B meson mass, and to reduce the systematic errors associated with
the extrapolation in terms of a heavy quark expansion. For decays of D and Dsmesons,
our results for the physical form factors at q2= 0 are as follows: fD→π
fD→K
+
(0) = 0.78(5)(4) and fDs→K
+
(0) = 0.68(4)(3). Similarly, for B and Bs we find:
fB→π
+
(0) = 0.27(7)(5), fB→K
+
(0) = 0.32(6)(6) and fBs→K
our results with other quenched and unquenched lattice calculations, as well as with light-
cone sum rule predictions, finding good agreement.
+
(0) = 0.74(6)(4),
+
(0) = 0.23(5)(4). We compare
PACS numbers: 11.15.Ha, 12.38.Gc, 13.20.Fc, 13.20.He
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1Introduction
Heavy meson decays are the main source of precision information on quark flavor mixing pa-
rameters in the Standard Model. The over-determination of the sides and the angles of the
CKM unitarity triangle is the aim of an extensive experimental study: It addresses the question
whether there is New Physics in flavor-changing processes and where it manifests itself. One of
the sides of the unitarity triangle is given by the ratio |Vub/Vcb|. Vcbis known to approximately
2% accuracy from b → cℓνℓtransitions [1, 2] whereas the present error on Vubis much larger
and there is also some tension between the determinations from inclusive and exclusive decay
channels. Reduction of this error requires more experimental statistics but—even more so—an
improvement of the theoretical prediction of the semileptonic spectra and decay widths.
This is the prime motivation for the study of semileptonic form factors of decays of a heavy
meson H = B,D into a light pseudoscalar meson P = π,K, which are usually defined as
?P(p)|Vµ|H(pH)? =m2
H− m2
q2
P
qµf0(q2) +
?
pµ
H+ pµ−m2
H− m2
q2
P
qµ
?
f+(q2).(1)
Here Vµ= q2γµq1is the vector current in which q1(q2) denotes a light (heavy) quark field; p
(pH) is the momentum of the light (heavy) meson with mass mP (mH), and q := pH− p is the
four-momentum transfer. The f0(q2) and f+(q2) form factors are dimensionless, real functions
of q2(in the physical region), which encode the strong interaction effects. They are subject to
the kinematic constraint f+(0) = f0(0).
In the approximation of massless leptons (which is highly accurate for ℓ = e or ℓ = µ), the
differential decay rate for the H → Pℓνℓprocess involves f+(q2) only:
dΓ
dq2=G2
192π3m3
H
F|Vq2q1|2
??m2
H+ m2
P− q2?2− 4m2
Hm2
P
?3/2 ??f+(q2)??2.(2)
Another motivation for our study is that f0(q2) and f+(q2) enter as ingredients in the analysis
of nonleptonic two-body decays like B → ππ and B → πK in the framework of QCD factor-
ization [3, 4], with the objective to extract CP-violating effects and in particular the angle α
of the CKM triangle. One issue that is especially important in this respect is the question of
flavor SU(3) violation in the form factors of the decay B → π vs. the rare decay B → K.
High-statistics unquenched lattice calculations of D-meson (and also B-meson) decay form
factors in the kinematic region where the outgoing light hadron carries little energy (small
recoil region) have been performed recently [5, 6, 7] and attracted a lot of attention. Direct
simulations at large recoil, q2≪ m2
2 GeV, prove to be difficult and require a very fine lattice which is so far not accessible in
calculations with dynamical fermions. This problem is aggravated by the challenge to consider
heavy quarks which either calls for using effective heavy quark theory methods or, again, a
very fine lattice. In practice, one is forced to rely on extrapolations from larger momentum
transfer q2and/or smaller heavy quark masses. Several extrapolation procedures have been
suggested [8, 9, 10, 11, 12] that incorporate constraints from unitarity and the scaling laws in
the heavy quark limit. Alternatively, B-meson form factors in the region of large recoil have
been estimated using light-cone sum rules [13, 14] (for recent updates see [15, 16, 17, 18]).
In this paper we report on a quenched calculation of semileptonic H → Pℓνℓform factors
with lattice spacing a ∼ 0.04 fm using nonperturbatively O(a) improved Wilson fermions and
B, with light hadrons carrying large momentum of order
1
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O(a) improved currents. On such a fine lattice a relativistic treatment of the c quark is justified
and also the extrapolation to the physical b quark mass becomes much more reliable compared
to similar calculations on coarser lattices. In addition, we can explore possible subtleties in
approaching the continuum limit in form factor calculations: In our previous work [19] we did
find indications for a substantial discretization error in the decay constants fDsetc.; similar
conclusions have also been reached in Ref. [20]. This is particularly relevant in view of the
claims of evidence for New Physics from comparison with recent dynamical simulations—see,
e.g. Ref. [21] for a discussion.
On physical grounds, one may expect a nontrivial continuum limit because form factors at
large momentum transfer are determined by the overlap of very specific kinematic regions in
hadron wave functions (either soft end-point, or small transverse separation). The common
wisdom that hadron structure is very “smooth”—and that numerical simulations on a coarse
lattice could thus be sufficient to capture the continuum physics—may not work in this par-
ticular case. This can be especially important for SU(3) flavor-violating effects, which are of
major interest for the phenomenology. Inclusion of dynamical fermions and the approach to the
chiral limit are certainly also relevant problems, but not all issues can be addressed presently
within one calculation.
This work should be viewed as a direct extension of the investigation of the APE collaboration
in Ref. [22], who performed a quenched calculation with the same nonperturbatively O(a)
improved action and currents. Also their data analysis is similar. However, they use coarser
lattices with a ∼ 0.07 fm (β = 6.2). On the other hand, the spatial volume of their lattices is
very close to ours (L ∼ 1.7 fm). So the main difference lies in the lattice spacing, and a direct
comparison of the results is possible yielding information on the size of lattice artefacts, while
there is no need for us to perform simulations on a coarser lattice ourselves.
The presentation is organized as follows. Our strategy is discussed in detail in Sec. 2. It
allows us to run fully relativistic simulations for values of mHup to the vicinity of mB: This is
achieved by using a lattice characterized by a very fine spacing a. The extraction of physical
quantities from our data and the final results with the associated error budget are presented
in Sec. 3. The final Sec. 4 contains a summary and some concluding remarks. Some technical
details and intermediate results of our calculation are shown in the Appendix. Preliminary
results of this study have been presented in Refs. [23, 24].
2Simulation details
The lattice study of heavy hadrons is an issue that involves some delicate technical aspects:
The origin of the problem stems from the fact that, typically, the lattice cutoff is (much) smaller
than the mass of the B meson.
Common strategies to solve this problem are based on heavy quark effective theory (HQET),
i.e. expanding the relativistic theory in terms of m−1
One can simulate in the static limit [25] or keep correction terms in the action to simulate
at finite mQ(non-relativistic QCD or NRQCD) [26]. These approaches have been employed
effectively for studying B physics (see, for example, Refs. [27, 28, 29, 30]).
However, for smaller quark masses like the c quark in D mesons a large number of terms in
the expansion must be included, making the simulations less attractive. The Fermilab group
Q, where mQis the mass of the heavy quark.
2
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developed an approach which interpolates between the heavy- and light-quark regimes [31].
The coefficients accompanying each term in the action are functions of the quark mass and
in practice, normally, the lowest-level action is used.
improved relativistic action with a re-interpretation of the results. Except for HQET [32, 33],
the associated renormalization constants for these approaches are only known perturbatively.
We reduce the uncertainties related to the extrapolation to the physical heavy meson mass
by using lattices with a small spacing in conjunction with a non-perturbatively improved O(a)
relativistic quark action. This theoretically clean approach enables one to get sufficiently close
to the mass of the physical B meson, so that the heavy-quark extrapolation is short-ranged.
In addition, in the region of the D meson mass, the discretization errors are reduced to around
1%, see Section 3.1.
This corresponds to using the O(a)
L3× T
β
403× 80
6.6
0.04 fm
1.63× 3.2 fm4
4.97 GeV
114
0.135472(11)
0.13, 0.129, 0.121, 0.115
0.13519, 0.13498, 0.13472
526 MeV, 690 MeV, 856 MeV
1.467
0.8118
1.356
−0.0874
lattice spacing a
physical hypervolume
a−1
# of configurations
κcritical
κheavy
κlight
mP
cSW
ZV
bV
cV
Table 1: Parameters of the lattice calculation (see the text for the definition of the various
quantities).
Table 1 summarizes basic technical information about our study. We use the standard Wilson
gauge action to generate quenched configurations with the coupling parameter β = 6.6. For this
parameter choice, the lattice spacing in physical units determined from Ref. [34] using Sommer’s
parameter r0= 0.5 fm is a = 0.04 fm. Our calculation is based on the O(a) improved clover
formulation for the quark fields [35], with the nonperturbative value of the clover coefficient cSW
taken from Ref. [36]. We use O(a) improved definitions of the vector currents in the form [37]
Vµ= ZV
?
1 + bVamq2+ amq1
2
?
(q2γµq1+ iacV∂νq2σµνq1)(3)
with σµν =
as cV are known nonperturbatively [38, 39, 40]. All statistical errors are evaluated through
a bootstrap procedure with 500 bootstrap samples. We consider three hopping parameters
corresponding to “light” quarks, κlight (the corresponding masses of the light pseudoscalar
meson states mPare also given in Table 1), and four hopping parameters, κheavy, corresponding
i
2[γµ,γν]. The renormalization factor ZV, the improvement coefficient bV as well
3
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p
H
decay product
quark
boson
W
lepton
neutrino
spectator quark
p
q
heavy quark
t t = T/2
yx
H
Vµ
0
PS
S
Figure 1: Diagram representing the semileptonic decay of a heavy-light pseudoscalar meson
to a light pseudoscalar meson (left panel). A schematic representation of the corresponding
three-point correlator calculated on the lattice is also shown (right panel).
to “heavy” quarks; in particular, κ = 0.13498 and κ = 0.129 are found to correspond to quark
masses close to the physical strange and charm quark mass, respectively.
The extraction of the matrix element appearing in Eq. (1) from the lattice can be done
by considering the large time behavior of three-point correlation functions C(3)
pseudoscalar light meson sink at time t = 0, a vector current at time tx, and a pseudoscalar
heavy-light meson source at time ty= T/2 (see Fig. 1):
µ (0,tx,ty) for a
C(3)
µ(0,tx,ty) =
?
? x,? y
e−i? pH·? yei? q·? x?HS(? y,ty)Vµ(? x,tx)PS(0)? .(4)
Here, HSand PSare Jacobi-smeared operators of the form qhγ5qsand qlγ5qs, respectively; qh
denotes the heavy quark, qlis the decay-product quark, while qsis the “spectator” quark.
For sufficiently large time separations (i.e. 0 ≪ tx≪ T/2 or T/2 ≪ tx≪ T), C(3)
behaves as:
with ZS
light (heavy) meson. To extract the matrix elements we divide the three-point functions by the
prefactors, which are extracted from fits to smeared-smeared two-point functions. The matrix
element is then obtained by fitting this result to a constant, in an appropriate time range where
a clear plateau forms (for example, for 12 ≤ tx≤ 28).
We consider three-point functions associated with different combinations of the momenta p
and pH, which are listed in Table 2. In particular, we focus our attention onto three-momenta
of modulus 0 and 1 [in units of 2π/(aL)], since they yield the most precise signal, restricting
ourselves to the cases where ? p and ? pH lie in the same direction. Thus we measure directly
5 different values for the form factors, for every κlightand κheavycombination. The full form
factors can then be constructed from the data points obtained this way, by making an ansatz
for the functional form of f0(q2) and f+(q2).
In the present work, we fit our data with the parametrization proposed by Be´ cirevi´ c and
Kaidalov [8]:
f0(q2) =cBK· (1 − α)
1 − ˜ q2/β
µ (0,tx,ty)
C(3)
µ(0,tx,ty) −→
ZS
4EHEe−Etxe−EH(ty−tx)?H(pH)|Vµ|P(p)?
HZS
for tx< T/2
±ZS
HZS
4EHEe−E(T−tx)e−EH(tx−ty)?H(pH)|Vµ|P(p)? for tx> T/2
H= |?0|HS|H(pH)?| and ZS= |?0|PS|P(p)?|, while E (EH) denotes the energy of the
,(5)
,f+(q2) =
cBK· (1 − α)
(1 − ˜ q2)(1 − α˜ q2),(6)
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? pH
? p? q
(0,0,0)
(1,0,0)
(0,0,0)
(1,0,0)
(1,0,0)
(1,0,0)
(−1,0,0)
(0,0,0)
(0,0,0)
(1,0,0)
(−1,0,0)
(2,0,0)
(0,0,0)
(1,0,0)
(0,0,0)
Table 2: Momentum combinations considered in the analysis of the three-point functions, in
units of 2π/(aL).
where ˜ q := q/mH⋆, mH⋆ being the mass of the lightest heavy-light vector meson.
The parametrization for the form factors given in Eq. (6) accounts for the basic properties
that come from the heavy-quark scaling laws in the limits of large and small recoil and also
satisfies the proportionality relation derived in Ref. [41]. It is also consistent with the trivial
requirement that the l.h.s. of Eq. (1) be finite for vanishing momentum transfer, which implies
f0(0) = f+(0). The results that we obtained for the three parameters entering Eq. (6) from a
simultaneous fit to f0and f+are presented in the Appendix.
Some alternative ans¨ atze for the functional form of f+(q2) were proposed in Refs. [15, 9, 10]
and are discussed in Ref. [42]: They yield results essentially compatible with each other and
with the Be´ cirevi´ c-Kaidalov parametrization Eq. (6). More recently, Bourrely, Caprini and
Lellouch [12] discussed the representation of f+(q2) as a (truncated) power series in terms of
an auxiliary variable z. A similar parametrization has also been recently used by the Fermilab
Lattice and MILC collaborations, see Refs. [43, 44] for a discussion.
3 Extraction of physical results
In order to extract physical results from our simulations, we follow a method analogous to
Ref. [22]. We first perform a chiral extrapolation in the light quark masses. For a given
quantity Φ [one of the BK parameters appearing in Eq. (6)], the extrapolation relevant for
decays to a pion is performed as follows: We fit the results obtained at different values of the
mass of the pseudoscalar state linearly in m2
P,
Φ = c0+ c1· m2
P, (7)
and extrapolate to m2
extrapolations are shown in Figure 2 for the case of Φ = f+(0), α and β at κheavy= 0.115. On
the other hand, for decays to a kaon, we hold the hopping parameter of one of the two final
quarks fixed to κ = 0.13498, which, for our configurations, corresponds to the physical strange
quark at a high level of precision [19], and perform a short-ranged extrapolation of the curve
obtained from the linear fit in m2
Then we perform the interpolation to the physical c quark mass in terms of a heavy-quark
expansion for the D (or Ds) meson decays, or the extrapolation to the physical b quark mass
for the B (or Bs) meson. For our data, the extrapolation of the heavy quark mass to the
physical b mass is short-ranged: for the heaviest κheavy= 0.115, it turns out that the inverse
of the pseudoscalar meson mass (with the light quark mass already chirally extrapolated to its
P= m2
π, where mπ is the mass of the physical pion. Examples of the
Pto the square of the mass of the physical K meson.
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00.010.02 0.030.04
0.05
(amP)2
0
0.2
0.4
0.6
0.8
1
f+(0)
00.010.020.030.04
0.05
(amP)2
0
0.5
1
1.5
α
00.010.020.030.04
0.05
(amP)2
0
0.5
1
1.5
2
2.5
3
β
Figure 2: Extrapolation of the Be´ cirevi´ c–Kaidalov parameters to the chiral limit, for decays to
a pion, at a fixed value κheavy= 0.115. The parameters obtained for κdecay product= κspectator
are extrapolated linearly in m2
P. The extrapolated values are shown as the full black dots.
Decay
B,D → π
fitl0
l1
l2
–
χ2/d.o.f.
0.1377/2
0.021/1
0.3247/2
0.03813/1
0.4025/2
0.001888/1
linear
quadratic
linear
quadratic
linear
quadratic
4.1+1.3
−1.0
5.1+2.9
−2.1
4.9+1.1
−0.9
6.3+2.4
−1.9
3.4+0.9
−0.8
4.9+1.8
−1.3
−4.1+1.6
−9.3+6.9
−5.4+1.5
−12.2+6.3
−2.9+1.3
−11.0+4.6
−2.3
−9.6
5.9+8.5
−5.9
–
7.7+7.3
−5.4
–
9.7+5.4
−4.6
B,D → K
−1.9
−8.2
Bs,Ds→ K
−1.7
−6.0
Table 3: The coefficients obtained from the fits to m3/2
Eq. (8) for different decays.
Hf+(0) in powers of m−1
Haccording to
physical value) is about m−1
physical B meson. The extrapolation can be performed by taking advantage of the fact that, in
the infinitely heavy quark limit, the Be´ cirevi´ c–Kaidalov parameters appearing in Eq. (6) enjoy
certain scaling relations: cBK√mH, (β−1)mHand (1−α)mHare expected to become constant
in the mH→ ∞ limit. For finite mH, one can parametrize the scaling deviations in powers of
m−1
ϕ = l0+ l1· m−1
where ϕ ∈ {cBK√mH, (β −1)·mH, (1−α)·mH}. Note that, since f+(0) = cBK·(1−α), one
can also use ϕ = f+(0) · m3/2
The extrapolation of m3/2
vantage of simulating on a fine lattice, which allows us to probe a mass range very close to
the physical B meson mass. We compare the results obtained from an extrapolation to the
inverse of the physical B meson mass using either a first- or a second-order polynomial in m−1
for the fit function, finding consistency (within error bars), for all decays. The corresponding
fit results are listed in Table 3. In the following we refer to this first method as the “coefficient
extrapolation” method.
An alternative method to extract the physical form factors from the lattice data was pro-
posed by the UKQCD collaboration [45]. It consists of performing the chiral and heavy quark
extrapolations at fixed v·p = (m2
meson and p is the four-momentum of the light meson. The following steps are performed:
H= 0.243 GeV−1, to be compared with m−1
B= 0.189 GeV−1for the
H:
H+ l2· m−2
H+ ...(8)
H—which was, in fact, our choice.
Hf+(0) is presented in Figure 3. The figure clearly shows the ad-
H
H+m2
P−q2)/(2mH), where v is the four-velocity of the heavy
1. fit of the form factors measured from the lattice simulations to the parametrization in
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00.1 0.20.3
-1 [GeV-1]
0.4
0.50.6
0.7
mH
0
1
2
3
4
5
mH
3/2 f+(0) [GeV3/2]
B, D to π decays
00.10.20.3
-1 [GeV-1]
0.4
0.50.6
0.7
mH
0
1
2
3
4
mH
3/2 f+(0) [GeV3/2]
Bs, Ds to K decays
Figure 3: Left panel: The green (red) squares denote the interpolated (extrapolated) form
factor m3/2
(solid line). A quadratic fit to the data is also shown (dashed line). Right panel: the results
for the case of decay of a Ds(Bs) meson into a kaon.
Hf+(0) to the physical D (B) meson, for a decay to a pion, using a linear fit in 1/mH
Eq. (6);
2. interpolation of the form factors at given values of v · p within the range of simulated
data;
3. chiral extrapolation of the points thus obtained, via a linear extrapolation in m2
m2
Pto either
πor m2
K(as described above);
4. linear or quadratic extrapolation in m−1
for the quantities:
Hto the inverse of the physical heavy meson mass
?αs(mB)
αs(mH)
?−
˜ γ0
2β0f0(v · p)√mH,
?αs(mB)
αs(mH)
?−
˜ γ0
2β0 f+(v · p)
√mH
(9)
which enjoy scaling relations at fixed v·p [46, 47]. Here, β0is the first β-function coefficient,
while ˜ γ0= −4 denotes the leading-order coefficient of the anomalous dimension for the
vector current in HQET. It yields a (subleading) logarithmic dependence on mH—see
also Refs. [22, 45] for further details;
5. final fit of the points thus obtained to the parameterization in Eq. (6).
For comparison, we also calculate the physical form factors using this alternative approach,
finding consistent results. This is illustrated in Table 4 which summarizes the results for f+(0)
from both methods.
For our final results we take those obtained from the coefficient extrapolation method. We
found this method to be superior in our case as the UKQCD method suffered from the fact that
there was only a small region of overlap in the ranges of v · p for the form factors at different
7
Page 9
κlightand κheavy. In addition, since the data can be fitted with both a linear and quadratic
function in m−1
differences in the results from the linear and quadratic fit to estimate the systematic errors, as
discussed in the next section. Our results for the form factors at finite q2are shown in Figs. 4
and 5.
H, we use the linear fits for the central values and statistical errors and use the
Coefficient extrapolation
linear in m−1
H
0.74+6
−6
0.78+5
−5
0.68+4
−4
0.27+8
−6
0.32+6
−5
0.23+5
−4
UKQCD method
linear in m−1
H
0.69+5
−5
0.75+4
−5
0.68+4
−4
0.29+13
−8
0.35+11
−8
0.23+6
−5
Decay
D → π
D → K
Ds→ K
B → π
B → K
Bs→ K
Final results for the physical values of the f+(0) form factor, for different decays,
with statistical errors only. We compare the results obtained from the coefficient extrapolation
and UKQCD methods as well as different truncations of the heavy quark expansion when
extrapolating or interpolating in m−1
quadratic in m−1
0.73+5
0.77+5
0.67+4
0.30+11
0.35+9
0.26+7
H
quadratic in m−1
0.69+5
0.75+4
0.67+4
0.31+15
0.34+12
0.27+8
H
−6
−6
−5
−5
−4
−4
−8
−10
−8
−9
−5
−6
Table 4:
H.
3.1Systematic uncertainties
Systematic uncertainties affecting our lattice calculation include: the quenched approximation,
the method to set the quark masses, the chiral extrapolation for the light quarks, discretization
effects, the extrapolation (interpolation) of the heavy quark to the physical b (c) mass, finite
volume effects, uncertainties in the renormalization coefficients, and effects related to the model
dependence for f0,+(q2). Let us now consider each source of error in turn.
Quenched approximation: the size of the error this approximation introduces is not
known. However, one can take as an estimate the variation in the results if different quantities
are used to set the scale. In the quenched approximation different determinations of the lattice
spacing vary by approximately 10% [48]. By repeating the full analysis, we find that varying
the lattice spacing by 10% induces an uncertainty of approximately 2% for the D → π decay,
and of approximately 12% for B → π.
Setting the quark masses: we use the κ values corresponding to the light (u/d) and
strange quarks determined in Ref. [19]: κl= 0.135456(10) and κs= 0.134981(9). The uncer-
tainty in these determinations leads to a very small uncertainty in the form factors. For the
c and b quarks we do not quote the corresponding κ values. We interpolate (or extrapolate)
our results directly to the physical masses of the pseudoscalar heavy-light states. The result-
ing uncertainty is determined by the statistical errors of the masses used for the interpolation
(or extrapolation). The latter are found to contribute only a negligible amount to the overall
systematic uncertainty.
Chiral extrapolation: the method we used to perform the chiral extrapolation of our
simulation results is discussed above. Note that the use of a large lattice practically constrains
us to use only a few and relatively large values for the light quark mass (so that the masses
of our lightest pseudoscalar mesons are far from the physical pion mass). However, as the
8
Page 10
-2 -1.5 -1 -0.50
0.5
1
1.5
2
2.5
q2 [GeV2]
0.5
1
1.5
2
f+
f0
D → π
QCDSF (this work)
Abada et al. [22]
Aubin et al. [5]
-2 -1.5 -1 -0.50
0.5
1
1.5
2
2.5
q2 [GeV2]
0.5
1
1.5
2
f+
f0
D → K
QCDSF (this work)
Abada et al. [22]
Aubin et al. [5]
-2 -1.5 -1 -0.50
0.5
1
1.5
2
2.5
q2 [GeV2]
0.5
1
1.5
2
f+
f0
Ds→ K
Figure 4: Physical form factors for D and Dsdecays as a function of q2from this work and
other quenched and dynamical studies. The solid black lines are the form factors obtained from
the coefficient extrapolation method where Eq. (8) has been truncated at O(m−1
dashed black lines indicate the error on the form factors. The range of v · p values achieved in
our simulations approximately corresponds to −1.5 GeV2? q2? 2 GeV2. The dashed red lines
are the results for the coefficient extrapolation method from Ref. [22]. The open red squares
and circles are their results obtained using the UKQCD method.
H), while the
examples in Figure 2 show, the dependence of our results on the light quark mass is rather
mild. So the size of the uncertainty arising from the chiral extrapolation though difficult to
estimate is unlikely to be large.
Discretization effects: as it was already remarked above, the leading discretization effects
in our calculation are reduced to O(a2); given that our lattice is very fine (a = 0.04 fm), the
associated systematic error can be estimated to be of the order of 1% (10%) for the decays of
charmed (beautiful) mesons [19].
Extrapolation/interpolation of the heavy quark: our data can be fitted to both a
linear and quadratic function in m−1
fit for our final results and the difference between the linear and quadratic fit as an indication
of the systematic uncertainty. This leads to approximately a 1% uncertainty for D decays and
Hwith a reasonable χ2. We use the results for the linear
9
Page 11
0
5
10
q2 [GeV2]
15
20
25
0
0.5
1
1.5
2
2.5
3
3.5
4
f+
f0
B → π
QCDSF (this work)
Abada et al. [22]
UKQCD [45]
Fermilab and MILC [44]
HPQCD [7]
0
5
10
q2 [GeV2]
15
20
25
0
0.5
1
1.5
2
2.5
3
3.5
4
f+
f0
B → K
0
5
10
q2 [GeV2]
15
20
25
0
0.5
1
1.5
2
2.5
3
3.5
4
f+
f0
Bs→ K
Figure 5: Same as in Fig. 4, but for B and Bs decays; in this case, the v · p values of our
simulations are in the range 14 GeV2? q2? 23 GeV2. For B → π, the dashed and solid
magenta lines in the range q2= 0 − 14 GeV2indicate the prediction from light-cone sum
rules [15, 16].
8% uncertainty for B decays.
Finite volume effects: for our calculation, finite-volume effects are not expected to be
severe; in particular, the correlation length associated with the lightest pseudoscalar state that
we simulated (for κlight= 0.13519) corresponds to approximately 9 lattice spacings, which is
more than four times shorter than the spatial extent of our lattice. Systematic infrared effects
can thus be quantified around 1–2%. This is comparable with the estimate of Ref. [44], in
which, using chiral perturbation theory [49, 50], the finite volume effects for their calculation
with 2 + 1 flavors of staggered quarks and values of mPL between 4 and 6 are estimated to be
less than 1%.
Renormalization coefficients: the uncertainty associated with the ZV coefficient, as de-
termined in Ref. [38] for the quenched case, is about 0.5%. The same article also quotes a
10
Page 12
1% uncertainty for bV, which induces an error about 1% for decays of D mesons and about
3% for B mesons. Concerning cV, a look at the results displayed in Fig. 2 of Ref. [39] would
suggest that the relative error in the region of interest (g2
30%; however, it should be noted that cV itself is a relatively small number, of the order of
9%, and the impact of the uncertainty on cV on our results is about 1% (2%) for decays of D
(respectively: B) mesons.
Model dependence: finally, the systematic effect related to the ansatz to parametrize the
form factors was estimated in Ref. [42], through a comparison of different functional forms that
satisfy analogous physical requirements. For the B → π decay, it turns out to be of the order
of 2%.
Combining the systematic errors in quadrature we arrive at an overall error of 5% for D
decays and about 18% for B decays.
0≃ 0.91) may be quite large, around
3.2Comparison with previous results
Our results can be compared to other lattice calculations of these quantities and also with
results of light-cone sum rules (LCSR) [13, 14]. Table 5 summarizes the comparison for f+(0),
while for finite q2the form factors from other studies are displayed in Figs. 4 and 5. In the
following we discuss in detail the comparison with these works, highlighting the advantages and
limitations of the different approaches, as well as the possible sources of discrepancies.
Our results can be closely compared with those obtained by the APE collaboration in
Ref. [22], reporting a calculation very similar to ours. They worked in the quenched approx-
imation, using the same non-pertubatively O(a) improved action and currents and a similar
analysis; on the other hand, their simulations were performed on a coarser lattice, with β = 6.2,
yielding a lattice spacing a = 0.07 fm, or a−1≃ 2.7 GeV. The table and figures show that their
values for the form factors lie around 3σ (D → π) and 2.5σ (D → K) below our results, in terms
of the statistical errors, in the region of q2= 0. If we adjust the APE results to be consistent
with setting the lattice spacing using r0instead of the mass of the K∗(used in Ref. [22]), the
discrepancy reduces slightly, down to roughly 2.5σ (D → π) and 2σ (D → K). Assuming that
O(a2) errors are the dominant source of the discrepancy, the difference in the results of the two
studies is consistent with an upper limit on the discretization errors of approximately 0.08, or
slightly above 1σ in our results for fD→π
+
(0) and 0.23 or 3 − 4σ in the APE results.
For B decays we are not able to make such a close comparison, because the study in Ref. [22]
extrapolates to the B meson from results in the region of 1.7 − 2.6 GeV for the heavy-light
pseudoscalar meson mass.Although one would expect larger discretization effects for the
B decay form factors, we find close agreement between our values and those from the APE
collaboration. However, we should point out that any potential discrepancy may be masked by
the long-ranged extrapolation in the heavy quark mass.
Several unquenched calculations have been performed recently, which are based on the MILC
Nf= 2+1 dynamical rooted staggered fermions configurations [51]. Results are available from
joint works from the Fermilab, MILC and HPQCD collaborations for D decays [5], and from
Fermilab and MILC [52, 44] and (separately) HPQCD [7] for B decays. These results were
obtained using the MILC “coarse” lattices with a = 0.12 fm for D decays and including a finer
lattice with a = 0.09 fm for the B decays. While these lattices are much coarser than those
used in both our and the APE study a detailed analysis of the chiral extrapolation was possible
11
Page 13
through the use of 5 light quark masses for the 0.12 fm lattice (only two values were used for
a = 0.09 fm).
The Fermilab, MILC and HPQCD joint work for D → π and D → K used an improved
staggered quark action (“Asqtad”) [51] for the light quarks and the Fermilab action for the
heavy quark. To the order implemented in the study, the Fermilab action corresponds to a re-
interpretation of the clover action. This approach can be used to simulate directly at the charm
and bottom quark mass at the expense of more complicated discretisation effects. Discretisation
errors arising from the final state energy (5%) and the heavy quark (7%) are estimated to lead
to the largest systematic uncertainties in the calculation (compared to the 3% error from the
chiral extrapolation). Given the coarseness of the lattice used, repeating the analysis on a
much finer lattice would enable the estimates of the the discretisation errors to be confirmed.
Overall, the results are consistent with ours, which suggests that the systematic effects due to
the quenched approximation are not the dominant source of error.
For the decay B → π, Fermilab and MILC used the same quark actions as for the study
of D decays. Using the 5 light quark masses at a = 0.12 fm and 2 light quark masses at
a = 0.09 fm they performed a joint continuum and chiral extrapolation which removed some
of the discretisation effects. They estimated that a 3% discretisation error arising from the
heavy quark remains after the extrapolation. The results at finite q2are compared with ours
in Figure 5, with statistical and chiral extrapolation errors only (which cannot be separated).
A value for f+(0) is not given in Ref. [44] which focuses on extracting |Vub| at finite q2using
the parameterisation of Bourrely, Caprini and Lellouch [12]. However, an earlier analysis on
the 0.12 fm lattices only was reported in Ref. [52]. Their result for f+(0) is given in Table 5.
HPQCD performed the calculations for the B → π decay on the MILC configurations using
Asqtad light quarks and NRQCD for the b quark. Use of the latter enables direct simulations
at the b quark mass. However, as NRQCD is an effective theory the continuum limit cannot
be taken and scaling in the lattice results must be demonstrated at finite a. Results from the
coarse lattice are shown in Figure 5, with statistical and chiral extrapolation errors only and for
f+(0) in Table 5. A limited comparison of results on the finer lattice for one light quark mass
did not indicate that the discretisation errors are large. The systematic errors are dominated
by the estimated 9% uncertainty in the renormalisation factors which are calculated to 2 loops
in perturbation theory.
The Fermilab-MILC and HPQCD results are consistent with each other to within 2σ and
are also consistent with our results and those of the APE collaboration. As for the studies of
D decays this suggests that quenching is not the dominant systematic error in the calculation
of B → π decay. Similarly, unquenched results on finer lattices are needed to investigate the
discretisation effects. Finally, note that in order not to overload Figures 4 and 5, we do not
show the (older) quenched results of the Fermilab group [53]. For B → π decays these results
are within the range of the other existing calculations, whereas for D-decays the form factors
come out 10 − 20% larger compared to most other calculations and also the new unquenched
results obtained with similar methods.
A different type of comparison can be made with the estimates obtained in the framework of
LCSR. This analytical approach is, to some extent, complementary to lattice calculations, since
it allows one to calculate the form factors directly at large recoil, albeit with some assumptions.
Figure 5 compares our extrapolation of the fB→π
+
with the direct LCSR calculation [15, 16]. Their predictions are compatible with our results.
(q2) form factor in the region q2< 12 GeV2
12
Page 14
Similar consistency is found between lattice and LCSR calculations of f+(0), as seen in Table 5,
for both B and D decays. Note that the uncertainty quoted for f+(0) for B decays is smaller
than that for D meson decays, and comparable with the precision of the lattice results. However,
while LCSR provides a systematic approach for calculating these quantities it is by definition
approximate and the errors cannot be reduced below 10 − 15%, unlike the lattice approach,
which is systematically improveable.
Decay
D → π
This work
0.74(6)(4)
Other results
0.64(3)(6)
0.57(6)(1)
0.65(11)
0.63(11)
0.73(3)(7)
0.66(4)(1)
0.78(11)
0.75(12)
Source Method
Fermilab-MILC-HPQCD [5]
APE [22]
Khodjamirian et al. [54]
Ball [55]
Fermilab-MILC-HPQCD [5]
APE [22]
Khodjamirian et al. [54]
Ball [55]
Nf= 2+1 LQCD
Nf= 0 LQCD
LCSR
LCSR
Nf= 2+1 LQCD
Nf= 0 LQCD
LCSR
LCSR
D → K0.78(5)(4)
Ds→ K
B → π
0.68(4)(3)
0.27(7)(5)0.23(2)(3)
0.31(5)(4)
0.26(5)(4)
0.258(31)
0.26(4)
0.26(5)
0.331(41)
0.36(5)
0.33(8)
0.30(4)
Fermilab-MILC [52]
HPQCD [7]
APE [22]
Ball and Zwicky [15]
Duplanˇ ci´ c et al. [16]
Wu and Huang [18]
Ball and Zwicky [15]
Duplanˇ ci´ c et al. [17]
Wu and Huang [18]
Duplanˇ ci´ c et al. [17]
Nf= 2+1 LQCD
Nf= 2+1 LQCD
Nf= 0 LQCD
LCSR
LCSR
LCSR
LCSR
LCSR
LCSR
LCSR
B → K0.32(6)(6)
Bs→ K
Table 5: Comparison of the results for f+(0) of the present work with other calculations,
obtained from lattice QCD (LQCD) simulations or from light-cone sum rules (LCSR) by various
groups. Where two errors are quoted the first is statistical and the second is the combined
systematic errors.
0.23(5)(4)
4Conclusions
In this article we have presented a lattice QCD calculation of the form factors associated with
semileptonic decays of heavy mesons.
We have performed a quenched calculation on a very fine lattice with β = 6.6 (a = 0.04
fm), which allows us to treat the D meson decays in a fully relativistic setup, and to get
close to the region corresponding to the physical B meson mass. The importance of small
lattice spacings for heavy-quark simulations has recently become clear in the context of the
determination of fDs, the decay constant of the Dsmeson. In spite of O(a) improvement, a
continuum extrapolation linear in a2seems to be reliable only for lattice spacings below about
13
Page 15
0.07 fm in the quenched approximation [19, 20]. Depending on the particular improvement
condition, even a non-monotonous a dependence can appear on coarser lattices.
In this work we have investigated to which extent the systematic effects caused by lattice
discretization and long-ranged extrapolations to the physical heavy meson masses may influence
the results of different lattice calculations in which all other sources of systematic errors are
treated in a similar way. For these reasons, the results of our study can be directly compared
with those by the APE collaboration in Ref. [22], which reports a very similar calculation on a
coarser lattice at β = 6.2 (a ≃ 0.07 fm) with the same lattice action and currents. Adjusting
the APE results so that they comply with our procedure for setting the physical value of the
lattice spacing, we find quite large discrepancies of roughly 2.5σ (D → π) and 2σ (D → K). If
we assume that O(a2) errors are the dominant source of this effect, the difference in the results
of the two studies suggests an upper limit on the discretization errors of approximately 0.08 or
slightly above 1σ in our numbers for fD→π
+
(0) and 0.23 or 3 − 4σ in the APE results.
It is, however, to be noted that the interpretation of this difference as a mere discretization
error is somewhat more ambiguous than in the case of the decay constants considered in [19, 20],
because the momentum transfer q2adds another parameter that has to be adjusted before the
comparison can be attempted. The corresponding comparison for B decays can, in addition,
be undermined by the long-ranged extrapolations in the heavy quark mass and/or q2. These
results suggest that, for high-precision phenomenological applications, completely reliable rel-
ativistic lattice calculations of these form factors could require even finer spacings, and that,
for dynamical simulations at realistic pion masses, this goal might be difficult to achieve in the
near future. While we believe that the progress in computational power will eventually allow
one to realize this formidable task, it is fair to say that, for the moment, the less demanding
approaches which interpolate between the D meson scale and non-relativistic results provide a
valid alternative.
Finally, a few words are in order about the general perspective for calculations of the semilep-
tonic form factors of heavy mesons. Form factors of B decays at small values of the relativistic
momentum transfer q2involve a light meson with momentum up to 2.5 GeV in the final state
and are very difficult to calculate on the lattice, mainly because no lattice effective field theory
formulation is known for this kinematics that would allow for the consistent separation of the
large scales of the order of the heavy quark mass, as implemented in the Soft-Collinear Effective
Theory.
Thus one is left with two choices. The first one is to calculate the form factors at moderate
recoil (m2
extrapolate to large recoil (m2
of this approach is that the calculations can be performed on relatively coarse and thus not
very large (in lattice units) lattices. Therefore dynamical fermions may be included, high
statistical accuracy can be achieved as well as a better control over the chiral extrapolation.
The disadvantage is that a reliable extrapolation from the q2> 12−15 GeV2regime accessible
in this method to q2= 0 may be subtle. However, this problem may be alleviated by a promising
new approach, “moving NRQCD” [56], which formulates NRQCD in a reference frame where the
heavy quark is moving with a velocity v. By giving the B meson significant spatial momentum,
relatively low q2can be achieved for lower values of the final state momentum thus avoiding
large discretisation effects.
For the particular case of the B → π semileptonic decay the problem of simulating at
B− q2∼ O(mBΛQCD)) using, e.g., the HQET or NRQCD expansion and then to
B−q2∼ O(m2
B)) guided by the dispersion relations. The advantage
14
Page 16
large recoil can be avoided, at least in principle, since the shape of the form factor f+(q2)
can be extracted from the experimental data on the partial branching fraction in different q2
bins, see, e.g., Ref. [57]. The normalization can then be fixed by comparison to lattice data
in the q2∼ 10 − 20 GeV2range. This strategy (see Ref. [58] for a detailed discussion) is
indeed promising and may lead to a considerable improvement in the accuracy of the |Vub|
determination from exclusive B decays provided the combined data analysis using the full
statistics of the BaBar and Belle experiments (∼ 4·108 ¯BB pairs) becomes available. However,
for rare decays, such as B → K∗γ, B → K∗µ+µ−etc., which are likely to take the central stage
at LHCb and super-B factories, a similar strategy seems to be unfeasible.
The second choice are simulations with fully relativistic heavy quarks on very fine and large
(in lattice units) lattices. This procedure is presently bound to the quenched approximation,
but the benefit is that the extrapolation in the heavy quark mass and potentially also in q2is
of much shorter range. It goes without saying that the inclusion of dynamical fermions and
the approach to the chiral limit are also important problems, but presently it is impossible to
address all relevant issues within one calculation.
In our opinion both methods are justified and we have chosen the second option in this paper.
It turns out that our final results for, e.g., the B → π decays are consistent with determinations
based on dynamical simulations and LCSR. This may indicate that the quenching effects are
rather moderate. From our experience, the main problem that limits the usefulness of this
approach is the construction of sources for the light hadrons which yield a good overlap with
states of large momentum. It seems that the presently used sources are not good enough in
this respect. Improved sources have to be developed if a similar calculation is attempted on a
larger scale in the future.
Acknowledgements
We warmly thank Tommy Burch for collaboration in the early stages of this project. The
numerical calculations were performed on the Hitachi SR8000 at LRZ Munich. This work
was supported by DFG (Forschergruppe Gitter-Hadronen-Ph¨ anomenologie) and GSI. A.A.K.
thanks the DFG and “Berliner Programm zur F¨ orderung der Chancengleichheit f¨ ur Frauen
in Forschung und Lehre” for financial support. S.C. acknowledges financial support from the
Claussen-Simon-Foundation (Stifterverband f¨ ur die Deutsche Wissenschaft). M.P. gratefully
acknowledges support from the Alexander von Humboldt Stiftung/Foundation and from INFN.
The University of Regensburg hosts the Collaborative Research Center SFB/TR 55 “Hadron
Physics from Lattice QCD”.
A Simulation results
Fig. 6 shows a subset of the form factors f0(q2) and f+(q2) that we extracted from our sim-
ulations, for different combinations of the κ values for the heavy and spectator quarks, with
κdecay product= 0.13472. In Table 6 we present our results for the fits to the simulation data
with the Be´ cirevi´ c–Kaidalov parameterization [8] according to Eq. (6). This parameterization
uses the vector meson mass mH⋆. Our results for mH⋆ in lattice units are shown in Table 7.
15
Page 17
κdec. prod.
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13519
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13498
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
0.13472
κheavy
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
0.13
0.129
0.121
0.115
κspect.
0.13519
0.13519
0.13519
0.13519
0.13498
0.13498
0.13498
0.13498
0.13472
0.13472
0.13472
0.13472
0.13519
0.13519
0.13519
0.13519
0.13498
0.13498
0.13498
0.13498
0.13472
0.13472
0.13472
0.13472
0.13519
0.13519
0.13519
0.13519
0.13498
0.13498
0.13498
0.13498
0.13472
0.13472
0.13472
0.13472
cBK· (1 − α)
0.775+39
0.724+49
0.484+88
0.39+10
0.742+32
0.674+37
0.442+61
0.364+68
0.750+29
0.685+34
0.456+49
0.387+59
0.783+33
0.740+41
0.536+71
0.419+84
0.781+30
0.735+34
0.512+54
0.428+60
0.779+28
0.726+27
0.484+41
0.407+46
0.821+32
0.781+34
0.565+60
0.472+78
0.802+28
0.762+30
0.530+43
0.450+50
0.803+26
0.751+27
0.524+35
0.425+39
1/βαχ2/d.o.f.
0.77
1.08
0.66
0.45
1.35
1.51
0.71
0.36
1.05
0.99
0.39
0.21
0.87
1.20
0.78
0.61
1.03
1.24
0.56
0.36
1.18
1.03
0.48
0.54
0.82
0.97
0.80
0.69
0.86
1.15
0.68
0.59
1.01
0.80
0.79
1.07
−45
0.580+85
0.604+95
0.73+11
0.78+11
0.658+79
0.714+81
0.808+87
0.84+8
0.670+77
0.730+70
0.821+78
0.829+76
0.587+78
0.596+85
0.67+11
0.76+10
0.613+63
0.627+67
0.723+77
0.765+77
0.664+62
0.690+61
0.803+65
0.822+65
0.565+74
0.560+78
0.66+11
0.72+11
0.620+60
0.625+63
0.734+78
0.764+77
0.648+50
0.673+53
0.768+57
0.823+58
−79
0.01+11
0.09+12
0.49+15
0.66+13
0.229+88
0.337+91
0.65+11
0.76+9
0.338+75
0.437+76
0.709+86
0.787+78
0.065+94
0.12+10
0.44+14
0.64+12
0.200+67
0.256+71
0.557+92
0.69+9
0.325+62
0.386+62
0.675+73
0.766+70
0.053+80
0.094+90
0.42+14
0.59+13
0.197+62
0.244+68
0.549+95
0.665+89
0.290+52
0.353+58
0.616+69
0.749+65
−10
−47
−90
−12
−72
−12
−16
−8
−14
−16
−37
−75
−81
−35
−85
−91
−52
−96
−11
−52
−10
−11
−30
−71
−74
−29
−82
−77
−42
−89
−98
−46
−88
−98
−35
−68
−83
−38
−70
−9
−65
−12
−14
−67
−12
−15
−28
−61
−66
−29
−66
−72
−46
−91
−98
−47
−97
−11
−26
−61
−62
−27
−63
−65
−36
−72
−85
−38
−75
−83
−31
−61
−70
−33
−65
−77
−54
−10
−12
−62
−12
−13
−26
−59
−58
−27
−59
−65
−42
−76
−88
−46
−83
−92
−22
−53
−53
−23
−53
−59
−31
−66
−71
−33
−67
−75
Table 6: Results of the fits of the lattice data with Eq. (6). Note that at vanishing q2one has
f+(0) = f0(0), which is given by cBK· (1 − α) (fourth column of this Table).
The parameters f+(0), α and β are then extrapolated to the light-quark mass as described in
Section 3 and illustrated in Fig. 2. Finally, the parameters describing the physical form factors
are obtained through extrapolation of these values to the physical B meson mass (or through
interpolation to the physical D meson mass), according to the heavy-quark scaling laws, see
16
Page 18
κheavy
0.13
0.13
0.13
0.129
0.129
0.129
0.121
0.121
0.121
0.115
0.115
0.115
κlight
0.13519
0.13498
0.13472
0.13519
0.13498
0.13472
0.13519
0.13498
0.13472
0.13519
0.13498
0.13472
amH
amH⋆
0.3949+18
0.4017+16
0.4110+14
0.4300+17
0.4366+16
0.4458+14
0.6804+20
0.6868+17
0.6956+15
0.8460+20
0.8523+17
0.8611+15
0.3681+13
0.3762+13
0.3867+11
0.4060+13
0.4138+13
0.4240+12
0.6672+16
0.6743+14
0.6836+12
0.8369+17
0.8437+15
0.8527+13
−11
−16
−10
−14
−10
−14
−11
−15
−11
−15
−10
−14
−13
−17
−12
−16
−12
−14
−14
−18
−13
−16
−12
−14
Table 7: Masses of the heavy-light pseudoscalar and vector states, H and H⋆, respectively, in
lattice units, for the different (κheavy,κlight) combinations.
Fig. 7.
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-5
0
5
10
15
q2 [GeV2]
0
0.5
1
1.5
2
2.5
3
κdecay product = 0.13472, κspectator = 0.13519
-5
0
5
10
15
q2 [GeV2]
0
0.5
1
1.5
2
2.5
3
κdecay product = 0.13472, κspectator = 0.13498
-5
0
5
10
15
q2 [GeV2]
0
0.5
1
1.5
2
2.5
3
κdecay product = 0.13472, κspectator = 0.13472
Figure 6: Sample of form factors directly measured in our simulations.
show the results obtained for κdecay product= 0.13472, and for different values of κspectator. In
each plot, the results for f0(denoted by full symbols) and for f+(empty symbols) are plotted
against the square of the transferred momentum q2. The results for different values of κheavyare
displayed using different symbols: diamonds (κheavy= 0.13), squares (κheavy= 0.129), circles
(κheavy= 0.121) and triangles (κheavy= 0.115).
The three panels
21
Page 23
00.10.20.3
mH
0.4
0.5 0.6
0.7
-1 [GeV-1]
0
1
2
3
4
5
6
7
f+(0) mH
3/2 [GeV3/2]
00.10.20.3
mH
0.4
0.50.6
0.7
-1 [GeV-1]
0
1
2
3
4
5
(β-1) mH [GeV]
0 0.10.20.3
mH
0.4
0.50.6
0.7
-1 [GeV-1]
0
1
2
3
4
5
(1-α) mH [GeV]
Figure 7: Extrapolation and interpolation of the (chirally extrapolated) Be´ cirevi´ c–Kaidalov
parameters in the heavy quark mass. The plots show the results obtained for the combinations
f+(0) · m3/2
of a B (red squares) or of a D meson (green squares) to a kaon. The results are obtained using
linear fits in m−1
fits to quadratic order in m−1
H (l.h.s. panel), (β − 1)mH(central panel) and (1 − α)mH(r.h.s. panel), for decays
H(solid lines) according to Eq. (8); for comparison, the curves resulting from
H(dotted lines) are also shown.
22
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